Neural coding and
information geometry
Hiroyuki Nakahara
Lab for Integrated Theoretical Neuroscience
RIKEN Brain Science Institute
Outline
1. Overview
2. Hierarchical model of neural
interactions
Section I
Overview
Introduction
Intelligent behavior (e.g. higher, cognitive, adaptive)
Achieved by collective neural activities
Decision making and reward-oriented behavior
— Neural computations
-- “Computations by collective neural activities”
Introduction
Neural computation – computational neuroscience:
At heart, interactions of neural activities
1015 bits/sec (#synapse x 1 bit/sec)
Terry Sejnowski
Most experimental findings so far
-- mostly by approach of “isolation”
(e.g., lesion, fMRI and single-unit studies)
(Bartels, Zeki 04)
How to go beyond the ‘isolation’ approach?
Introduction
“modeling” – rather abstract form
e.g. Amari-Hopfield model, Boltzmann machine,
gradient disecent (natural ….), em-algorithm, graphical model, etc
“modeling” – neural coding / population coding
Tuning curve (spike counts), their variability
ri = φi ( x; ci ) + ni ( x)
ni ( x)
N (0,σ i2 ( x))
Q: About input x, given the activities r ?
e.g. Fisher information etc
Our past works
* Faithful and unfaithful models -- Wu et al. (’01, ’02a,b, ’04)
* Attention modulation -- Nakahara et al (’01, ’02)
* Singularity in decoding – Amari & Nakahara (’05)
* Neural dynamics in SC – Nakahara et al (’06)
Introduction
“data analysis”
“method of data analysis”
Spike coding (Neural spike as binary variable)
ηi = E[ xi ]
Most research with data so far: 1st-order or 2nd-order model
log Q pair ( X ) = ∑ θ i xi + ∑ θij xi x j −ψ
i
i, j
How can we reliably detect usefulness and meanings of
higher-than-pairwise order interactions of neural activities?
Information geometry on binary random vectors
Information Geometric Approach
Basic needs : Systematic treatment for higher-order interaction
Use good properties of the dual coordinates
log P ( X ) = ∑ θi xi + ∑ θij xi x j + ∑ θijk xi x j xk + K + θ12Kn x1 x2 K xn −ψ
i
ij
ijk
η -coordinates: η = (ηi , ηij , ...) = (η1 ,η2 ,...,ηN ),
ηA = E[ X A ]
θ -coordinates: θ = (θ i , θ ij ,...) = (θ1 ,θ 2 ,...,θ N )
k-cut mixed coord: ς k = (η1 , η2 ,...,ηk ;θ k +1 ,θ k + 2 ,...,θ N ) = (ηk- ;θ k + )
⎡ Aζ k
FI is given as Gζ k = ⎢
⎢⎣ O
O ⎤
⎡ Aη
⎥ , where Gη = ⎢ T
Dζ k ⎥⎦
⎣ Bη
Bη ⎤
,
Dη ⎥⎦
⎡ Aθ
Gθ = ⎢ T
⎣ Bθ
Aζ k = Aθ−1 ,
Pythagorian: D[ P : Q] = D[ P : R] + D[ R : Q] where ς kR = ( ηkP− ; ς Qk + )
Past works
Bθ ⎤
Dθ ⎥⎦
Dζ k = Dη−1
* IG framework for spike analysis -- Nakahara & Amari (‘02)
* Detection of interaction cascade -- Nakahara et al. (‘03)
* Emergent higher-order synchrony -- Amari et al. (‘03)
* Temporal domain — Nakahara et al (’06)
* Others etc….
Section II
“Theory in practice?”
Hierarchical Interaction Structure of Neural
Activities in Cortical Slice Cultures
(Santos, Gireesh, Plenz & Nakahara, J. Neurosci, 2010)
Introduction
• Previous studies suggested that 2nd-order maximum
entropy model (“pairwise model”) adequately explains
activity patterns.
log Q pair ( X ) = ∑ θ i xi + ∑ θij xi x j −ψ
i
i, j
• If generally true, a significant simplication for describing
neural interactions
Schneidman et al., 2006
Other works
(Shlens et al 06, 09, Tang et al 08 etc)
• Certainly, “appropriately simple” models are crucial for
improving our understanding of neural interactions and
thereby functions.
• But is only up to 2nd-order really sufficient for any
underlying network systems?
cf. for computations as well as scalability
• There may be other simplified models that correspond
better to an underlying network interaction structure,
thereby being adequate for large-scale cortical activity
---- what are appropriate units for interactions?
Our dataset (Plenz Lab, NIMH)
(hn note)
“neural avalanches”
power law
• Cultures of coronal slices from rat somatosensory cortex and the VTA
• LFP signal (1-200 Hz), thresholded to obtain negative LFP (nLFP) peaks
• nLFP peaks binned at 4, 10, or 20 ms
Pairwise model results
Pairwise model on a group of
10 electrodes (randomly-sampled)
Good results in 12 datasets
(200 groups / dataset)
The score called Fpair is frequently used :
Fpair = 1 -
(
( Pˆ ( X ) Q
)
( X))
D KL Pˆ ( X ) Q pair ( X )
D KL
ind
Hierarchical Model: Proposal
Our proposal - Define a new functional unit for large-scale activity
- Create a hierarchical model for different spatial scales
Intuition: low resolution of
correlation structure
COR of every electrode
pair may not be needed
Electrodes
as units
Clusters
as units
• How to define cluster activities?
• How to find clusters?
Cluster activity
= magnitude of electrode activity in cluster
Measures of cluster activity:
Linear
Examples
n
lin
I
= ∑ xi
log
I
= ⎢⎣log 2 (1 + cIlin ) ⎥⎦
c
i =1
Log
c
Binary
cIbin = ( cIlin > 0 )
Results shown later, mostly for log cluster activity
Clustering criterion: homogeneity of activity
A backward searching
procedure was used.
(note: caveat)
Hierarchical Model: Formulation
Hierarchical model integrates interactions over different scales
m
c
P ( X ) ≈ Qhier ( X ) = Q pair
(C) ∏
I =1
e
Q pair
( XI )
e
Q pair
( CI )
C = ( c1 , c2 ,K , cm )
• Pairwise models for both local (unit = electrode) & long-range (unit =
cluster) interactions, with conditional independence assumption on
electrode activity across clusters given cluster activity.
Results: clusters
Clusters - example array:
Homogeneous clusters characterized by
strong correlation and electrode proximity
within cluster
across clusters
• Physical distance on array
• Correlation coefficients
Results: 10-electrode
– comparison with pairwise model
One example of a 10 electrode group
Cluster information used
for 10-electrode groups
Example results for 1 group
(Fpair = 0.89, Fhier = 0.93)
Comparison with pairwise model (10-electrode groups)
Hierarchical model has better accuracy than
pairwise model, even with fewer parameters
Summary
(12 datasets, 200 groups / dataset)
Accuracy and # of param
(over 12 datasets)
Note: also confirmed with the results using cross-validation
also confirmed w.r.t. ‘shuffled’ clusters
Results: accuracy on full array (~ 60 electrodes)
Hierarchical model with log clusters
predicts array-wide synchronized states
Summary
Example
(one dataset; for a ‘fraction’)
P(f)
(with log – colored, binary -- gray)
Until here is in Santos et al., J. Neurosci, 2010
Ongoing results:
Getting back to interactions at electrode level
m
c
P ( X ) ≈ Qhier ( X ) = Q pair
(C) ∏
I =1
e
Q pair
( XI )
e
Q pair
( CI )
C = ( c1 , c2 ,K , cm )
Example
Original param.
e
Q pair
(xα ) :{θiα , θijα }
c
Q pair
(c) :{ζ α |a , ζ α ( β )α ′( β )|ab }
θ − coordinates (electrode level)
Q (xα ) :{θ% , θ% , θ% ,..., θ%
}
hier
i
ij
ijk
123...n
(Note: possible to write analytically)
Discussion
• The hierarchical model captures two levels of interactions
using two units: electrodes and clusters.
• The model captures cortical LFP activity patterns better than
the pairwise model. Æ clusters -- underlying cortical layer structure?
Æ Identifying the appropriate units of interaction of a network
may enable the network interactions to be better characterized,
with better pasimony and scalability
• Significant higher-order interactions are embedded in a
specific way. A new hierarchical model further helps us examine
those properties. cf. across/within cluster interactions
Follow-up (Q&A session)
• I make a note here regarding “context specific graphical
model” and/or “a weak definition of conditional independence”,
which I mentioned in Q & A session. Here is the paper I was
referring to; Nakahara et al. (2003) Bioinformatics. (you can also
download it from www.itn.brain.riken.jp). Any comments and
feedbacks will be greatly appreciated.
• Taking advantage of adding this slide after my presentation, I
would like to express my sincere gratitude to all the particiants
in the workshop and the organizers who have all of us get
together.
END